# cold_plasma_permittivity_LRP

plasmapy.formulary.dielectric.cold_plasma_permittivity_LRP(B: Quantity, species: , n: , omega: Quantity) [source]

Magnetized cold plasma dielectric permittivity tensor elements.

Elements (L, R, P) are given in the “rotating” basis, i.e. in the basis $$(\mathbf{u}_{+}, \mathbf{u}_{-}, \mathbf{u}_z)$$, where the tensor is diagonal and with $$B ∥ z$$.

The $$\exp(-i ω t)$$ time-harmonic convention is assumed.

Parameters:
Returns:

Notes

In the rotating frame defined by $$(\mathbf{u}_{+}, \mathbf{u}_{-}, \mathbf{u}_z)$$ with $$\mathbf{u}_{\pm}=(\mathbf{u}_x \pm \mathbf{u}_y)/\sqrt{2}$$, the dielectric tensor takes a diagonal form with elements L, R, P with:

\begin{align}\begin{aligned}L = 1 - \sum_s \frac{ω_{p,s}^2}{ω\left(ω - Ω_{c,s}\right)}\\R = 1 - \sum_s \frac{ω_{p,s}^2}{ω\left(ω + Ω_{c,s}\right)}\\P = 1 - \sum_s \frac{ω_{p,s}^2}{ω^2}\end{aligned}\end{align}

where $$ω_{p,s}$$ is the plasma frequency and $$Ω_{c,s}$$ is the signed version of the cyclotron frequency for the species $$s$$ .

Examples

>>> import astropy.units as u
>>> from numpy import pi
>>> B = 2 * u.T
>>> species = ["e-", "D+"]
>>> n = [1e18 * u.m**-3, 1e18 * u.m**-3]
>>> omega = 3.7e9 * (2 * pi) * (u.rad / u.s)
>>> L, R, P = permittivity = cold_plasma_permittivity_LRP(B, species, n, omega)
>>> L
<Quantity 0.63333...>
>>> permittivity.left  # namedtuple-style access
<Quantity 0.63333...>
>>> R
<Quantity 1.41512...>
>>> P
<Quantity -4.8903...>